A194002 Numbers k that are the start of a sequence of 7 maximally-squarefree numbers.

1, 65, 137, 209, 217, 281, 353, 433, 641, 713, 785, 793, 857, 937, 1001, 1217, 1289, 1361, 1433, 1505, 1577, 1657, 1793, 1865, 1937, 2081, 2089, 2233, 2305, 2377, 2441, 2513, 2585, 2665, 2729, 2801, 2953, 3017, 3089, 3161, 3241, 3305, 3313, 3457, 3529, 3593

Offset

1,2

Comments

k, k+1, k+2, k+4, k+5, and k+6 are squarefree; k+3 is divisible by 4 but no higher power of 2 and no other prime squared.

From Amiram Eldar, Nov 28 2023: (Start)

All the terms are of the form 8*k + 1.

The numbers of terms not exceeding 10^k for k = 1, 2, ... , are 1, 2, 14, 140, 1384, 13774, 137784, 1378053, 13779491, 137794128, 1377940943, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0137794... . (End)

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Mathematica

sfQ[n_]:=Module[{c4=FactorInteger[n[[4]]], r=Drop[n, {4}]}, First[c4] == {2, 2} && Max[Transpose[Rest[c4]][[2]]]==1&&And@@SquareFreeQ/@r]; Join[{1}, Transpose[ Select[Partition[Range[2, 3600], 7, 1], sfQ]][[1]]] (* Harvey P. Dale, Nov 22 2011 *)

Prog

(PARI) ap(n)={forstep(k=1, n, 8,
if(issquarefree(k)&&issquarefree(k+1)&&issquarefree(k+2)&&
   issquarefree((k+3)\2)&&
   issquarefree(k+4)&&issquarefree(k+5)&&issquarefree(k+6),
  print1(k", ")))}

Crossrefs

Subsequence of A005117, A007674, A007675 and A017077.

Sequence in context: A294169 A073631 A285300 * A092226 A211255 A297849

Adjacent sequences: A193999 A194000 A194001 * A194003 A194004 A194005

Keyword

nonn

Author

Franklin T. Adams-Watters, Aug 10 2011

Status

approved